2013/06/27 by Daniel El-Baz, El-Baz, Daniel, Jens Marklof +3 · 3 citations
Computer Science · Economics, Econometrics and Finance · Mathematics · #11J71 #11K36 #11P21 #22E40 #37A17 #37A25 #Advanced Mathematical Identities #Analytic Number Theory Research #Bayesian Methods and Mixture Models #Dynamical Systems (math.DS) #FOS: Mathematics #Financial Risk and Volatility Modeling #Mathematical Approximation and Integration #Mathematical Dynamics and Fractals #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1306.6543
openalex publication_date 2013/06/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Elkies and McMullen [Duke Math.J.~123 (2004) 95--139] have shown that the\ngaps between the fractional parts of \√ n for n=1,\…,N, have a limit\ndistribution as N tends to infinity. The limit distribution is non-standard and\ndiffers distinctly from the exponential distribution expected for independent,\nuniformly distributed random variables on the unit interval. We complement this\nresult by proving that the two-point correlation function of the above sequence\nconverges to a limit, which in fact coincides with the answer for independent\nrandom variables. We also establish the convergence of moments for the\nprobability of finding r points in a randomly shifted interval of size 1/N. The\nkey ingredient in the proofs is a non-divergence estimate for translates of\ncertain non-linear horocycles.\n